Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1971 A Feynman Path-Integral Calculation of the Polaron Effective Mass. Manmohan Singh Chawla Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Chawla, Manmohan Singh, "A Feynman Path-Integral Calculation of the Polaron Effective Mass." (1971). LSU Historical Dissertations and Theses. 1910. https://digitalcommons.lsu.edu/gradschool_disstheses/1910 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. 71-20,582 CHAWLA, Manmohan Singh, 1940- A FEYNMAN PATH-INTEGRAL CALCULATION OF THE POLARON EFFECTIVE MASS. The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1971 Physics, solid state ■i University Microfilms, A XEROX Company, Ann Arbor, Michigan I THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED A FEYNMAN PATH-INTEGRAL CALCULATION OF THE POLARON EFFECTIVE MASS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Physics and Astronomy by Manmohan Singh Chawla B.Sc., University of Rajasthan, 1960 M.Sc., University of Rajasthan, 1963 A. Inst. N.P., Calcutta University, 1965 January, 1971 PLEASE NOTE: Some pages have small and indistinct type. Filmed as received. University Microfilms ACKNOWLEDGEMENTS The author wishes to express hi's appreciation to Dr. John T. Marshall for his guidance and assistance at all stages of this research. Acknowledgement is also given here to the Louisiana State University Computer Center for use of its IBM 360/65 computer on which the numerical calculations were performed. Financial assistance from the Dr, Charles E. Coates Memorial Fund of Louisiana State University Foundation to partially defray the cost of preparation of this manuscript is also gratefully acknowledged. ABSTRACT In the expansion of the ground-state energy of a polaron in a weak magnetic field as a power series in the magnetic field strength, the zeroth-order term is the polaron self-energy while the first-order term is inversely proportional to the polaron effective mass. The effective mass so obtained is exactly equivalent to the free polaron effective mass as defined by Frohlich. This equivalence principle is used to approximate the polaron effective mass by employing an approximate expression of the ground- state energy of a polaron in a weak magnetic field obtained by applying Feynman's path-integral variational method. The resultant polaron effective mass is found to be higher than Feynman's result by less than one per cent. CONTENTS Chapters Page 1. INTRODUCTION .................. 1 2. EXACT EXPRESSION FOR THE POLARON GROUND- STATE ENERGY ................................ 10 3. CALCULATION OF THE POLARON EFFECTIVE MASS. 20 4. RESULTS AND CONCLUSION ...................... 32 Appendices A. FROHLICH'S POLARON MODEL ..................... 38 B. MARSHALL-ROBERT THEOREM ........... 57 C. VARIATIONAL PRINCIPLE ..... ‘............ 66 D. SOLUTION OF THE INTEGRO-DIFFERENTIAL EQUATIONS . 69 E. WEAK AND STRONG COUPLING EXPANSIONS.......... 76 F. NUMERICAL C A L C U L A T I O N S...................... 81 References............................. 95 Vita..................................... 97 CHAPTER I INTRODUCTION The polaron problem arises in the study of the motion of a single electron in the conduction band of a semi- conductive ionic crystal. The usual initial approach to the description of a conduction electron is the Bloch 1 approximation which assumes that the lattice ions are rigidly attached to their lattice sites. In the Bloch approximation an electron with energy near the bottom of the conduction band moves as a free particle with a certain mass m, called the band mass. For an ionic crystal this approximation is an over simplification because in such crystals the lattice ions are unshielded charges and there­ fore react significantly to the presence of a conduction electron. Thus the conduction electron induces an appre­ ciable displacement of the lattice ions from the static configuration assumed in the Bloch approximation. The electron in turn experiences not only the static field assumed in the Bloch approximation but also the incremental electric field resulting from the displacement of the lattice ions from their lattice sites. If one assumes that the wave vector of electron is small compared to the reciprocal lattice spacing {which is tantamount to saying that electronic wave function will vary only negligibly over the distance of several lattice spacings) then the electron can be regarded essentially as moving under the 1 2 influence of a continuous macroscopic polarization field arising from the displacement of ions from their lattice sites. The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying (non-radiative) polarization field. Another way to look at the motion of electron in an ionic crystal is to regard the electron as triggering quantized vibrations (phonons) in the crystal. Then the polaron is the Bloch electron "dressed" with its cloud of phonons. 2 In a derivation of the Hamiltonian for such a system, Frohlich invokes the additional approximation that the electron interacts predominantly with long wavelength longitudinal optical (LO) modes of the crystal all of which are treated as having a common frequency, w. Frohlich's A Hamiltonian is derived in Appendix A and is given by H(a> = | p2 + Z | (pj + q£ - 1) 1C ++ o rr x. -i cos k*r +j. [r _®v2rrOt _ ] r£ gk ' {r ^ ~ },i ii(l) \ k 1 ~ 1 ~ s m k*r where r is the electron coordinate vector, q^ is the polarization-field oscillator coordinate of wave vector k, p and p, are the momenta canonically conjugate to r and «w K n q^, and a is a dimensional constant called the polaron * Throughout this paper a natural system of units is used in which m=(D=v&=l. 3 coupling constant defined by wherein -e is the electronic charge and £ and em are the static and optical dielectric constants of crystal. In Eq. (1) V is a normalization volume which is to be taken infinitely large, in which limit Z + -Zr Jd3k. (3) k 8tt' The notation cos k*r { ~ ~ } (4) sin k*r ♦W , HI stands for the upper or the lower expression depending respectively on whether or not k satisfies one or the other of the following: k > 0, x ' k = 0. k > 0 , x y or k = 0, k = 0,k > 0, (5) x y z “The~'firstterm in th^' Hamiltonian given by Eq. (1) is the band energy of the electron measured from the bottom of the conduction band, the second term represents the energy of the polarization field measured relative to its zero-point energy, and the third term is the electron- IiO-phonon interaction energy. 2 Frohlich also defines the polaron self-energy, EQ (a), and the polaron effective mass, u(ct), by the expansion V “ '5r» “ Eo (a) + 2u W + ' 161 where E (a,k ) is the ground-state eigenvalue of H(a), O *** X whose corresponding eigenfunction, if* , , is also an °lKn eigenfunction of the total wave vector, Km , with eigen- value km. The total wave vector, K^, means the vector sum of the wave vectors of the electron and all the phonons. By definition, ip , depicts a situation in which all phonons present are non-radiative and are therefore part of the polaron's phonon cloud. Thus E O (a,krp)X represents the energy of a ground-state polaron propagating with a wave vector kT. The polaron problem has been of long standing interest in physics. From a theoretical viewpoint it is of interest because it affords a relatively simple but non trivial example of an interaction between a particle 5 and its self-produced quantized field in which no trouble­ some infinite renormalizations arise. Because of this extended interest, the coupling constant, a, is regarded as an independent parameter. One of the aims of the theoretical research is to develop calculational methods which are accurate for all values of the coupling constant, a, with the hope that such methods may be helpful in dealing with similar many-body problems such as a nucleon interacting with its meson field. Recent experimental investigations on polarons have further stimulated the theoretical work on the subject. First, the experiments on the ionization energy of bound polarons, 3 ' 4 require for their interpretation the theoretical difference in energy of the free-polaron ground-state energy and the energy of the bound polarons. Since these energies are usually calculated separately, it is important to have good values for the free-polaron ground-state energy. Second, polaron effective masses have been estimated experimentally by cyclotron resonance measurements made on a variety of materials in which the 5_o polaron coupling constant, a, lies in the range l<a<4.5. It would be desirable, therefore, to have a reasonably accurate formula for the polaron effective mass as a function of the polaron coupling constant (which in turn is known in terms of the band mass by Eq. (2)) because such a formula can be used in conjunction with the 6 'expe^inehtald.y “determined polaron mass to compute the band mass for the sake of comparison with independent theoretical calculations for the band mass. Various theoretical methods 9 ' 10 have been employed to approximate the polaron self-energy and effective mass. The Lee-Low-Pines"^ weak-coupling variational calculation yields Ec = -a , (7) and y(a) = 1 + g a . (8) These results are exactly correct to the first order in a in the limit of weak coupling.